Hermite finite elements for convection-diffusion equations

TL;DR

This paper develops Hermite finite element techniques for convection-diffusion equations, emphasizing flux continuity and comparing their performance to classical mixed methods through convergence analysis and numerical experiments.

Contribution

It introduces Hermite finite element methods for convection-diffusion problems, extending flux-continuous schemes and comparing them with Raviart-Thomas based methods.

Findings

Hermite methods show good flux continuity properties.

Numerical results demonstrate competitive accuracy.

Convergence analysis confirms theoretical predictions.

Abstract

This work addresses techniques to solve convection-diffusion problems based on Hermite interpolation. We extend to the case of these equations a Hermite finite element method providing flux continuity across inter-element boundaries, shown to be a well-adapted tool for simulating pure diffusion phenomena (cf. V. Ruas, J. Comput. Appl. Maths., 246 p. 234-242, 2013). We consider two methods that can be viewed as non trivial improved versions of the lowest order Raviart-Thomas mixed method, corresponding to its extensions to convection-diffusion problems proposed by Douglas and Roberts (cf. Computational and Applied Mathematics, 1, p. 91-103, 1982) . A detailed convergence study is carried out for one of the methods, and numerical results illustrate the performance of both of them, as compared to each other and to the corresponding mixed methods.